l, m, n are generally written as direction cosines or the direction ratios of unit vector-*,

Let l₁ = , m₁ = , n₁ =

l₂ = , m₂ = , n₂ =

l₃ = , m₃ = , n₃ =

For the lines or vectors to be perpendicular their dot product or scalar product should be zero and for the lines or vectors to be parallel their cross product or vector product should be zero.

So we will use scalar product to prove these lines perpendicular to each other.

l₁l₂ + m₁m₂ + n₁n₂ =

l₂l₃ + m₂m₃ + n₂n₃ =

l₁l₃ + m₁m₃ + n₁n₃ =

Therefore, all three lines or vectors are mutually perpendicular to each other.